Math Problem Statement

Given the following matrix B = [[4, 1], [2, 3]], find the eigenvalues of matrix B and the corresponding eigenvectors for each eigenvalue.

Solution

For the given matrix B=[4123]B = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}:

  1. The eigenvalues of matrix BB are:

    • λ1=5\lambda_1 = 5
    • λ2=2\lambda_2 = 2
  2. The corresponding eigenvectors for each eigenvalue are:

    • For λ1=5\lambda_1 = 5, the eigenvector is [11]\begin{bmatrix} 1 \\ 1 \end{bmatrix}
    • For λ2=2\lambda_2 = 2, the eigenvector is [121]\begin{bmatrix} -\frac{1}{2} \\ 1 \end{bmatrix}

Would you like more details or have any questions?

Here are 5 related questions you might explore:

  1. How do you verify the correctness of eigenvalues and eigenvectors?
  2. What is the significance of eigenvalues in practical applications?
  3. Can you calculate the determinant of matrix BB using its eigenvalues?
  4. What does it mean if a matrix has repeated eigenvalues?
  5. How do eigenvectors relate to diagonalization of a matrix?

Tip: Eigenvectors corresponding to distinct eigenvalues are always linearly independent.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Eigenvalues
Eigenvectors
Matrices

Formulas

det(B - λI) = 0 for eigenvalues
(B - λI)v = 0 for eigenvectors

Theorems

Eigenvalue Theorem
Determinant for Eigenvalues

Suitable Grade Level

Undergraduate - Linear Algebra